The Box Problem

by

Angie, Beth, and Teisha

Problem:
*Use the Arithmetic Mean-- Geometric Mean Inequality to find the maximum volume of a box made from a 25 by 25 square sheet of cardboard by removing a small square from each corner and folding up the sides to form a lidless box.


*Use the AM--GM Inequality to determine what shape boxes could be created by this method from the 25 by 25 square sheet to hold a volume of 100 cu. units? 200 cu. units? 400 cu. units?
*Generalize. Use the AM--GM Inequality to discuss the maximum volume of a box formed from an n X n square sheet of cardboard.
*Why will the AM--GM Inequality not be a useful tool when the sheet of cardboard is 20 by 25?


SOLUTION
The volume of our box is

To find the maximum volume, x, (25-2x), and (25-2x) must sum to a constant and they should be equal to one another. To meet these conditions, we need to multiply our volume equation by 4/4.
Our inequality is


To find the maximum volume,

Now, the maximum value occurs when x =25/6.

The maximum volume is 1156.41 cu. units.


V(x) = 100 cu units

The generalize formula for a n X n square.




For any n X n square, the maximum volume occurs when x=n/6.


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